We are given a set of elements in a metric space. The distribution of the elements is arbitrary, possibly adversarial. Can we weigh the elements in a way that is resistant to such (adversarial) manipulations? This problem arises in various contexts. For instance, the elements could represent data points, requiring robust domain adaptation. Alternatively, they might represent tasks to be aggregated into a benchmark; or questions about personal political opinions in voting advice applications. This article introduces a theoretical framework for dealing with such problems. We propose clone-proof weighting functions as a solution concept. These functions distribute importance across elements of a set such that similar objects (``clones'') share (some of) their weights, thus avoiding a potential bias introduced by their multiplicity. Our framework extends the maximum uncertainty principle to accommodate general metric spaces and includes a set of axioms -- symmetry, continuity, and clone-proofness -- that guide the construction of weighting functions. Finally, we address the existence of weighting functions satisfying our axioms in the significant case of Euclidean spaces and propose a general method for their construction.

翻译：给定度量空间中的一组元素。元素的分布是任意的，可能具有对抗性。我们能否以抵抗此类（对抗性）操纵的方式为元素分配权重？该问题出现在多种情境中。例如，元素可能代表数据点，需要鲁棒的领域自适应；或者它们可能代表需要聚合为基准的任务；亦或是投票建议应用中关于个人政治观点的问题。本文提出了处理此类问题的理论框架。我们引入克隆证明权重函数作为解决方案。这些函数在集合元素间分配重要性，使得相似对象（“克隆体”）共享其（部分）权重，从而避免因其多重性引入潜在偏差。我们的框架将最大不确定性原理推广至一般度量空间，并包含一组公理——对称性、连续性与克隆证明性——以指导权重函数的构建。最后，我们探讨了在欧几里得空间这一重要情形下满足公理的权重函数的存在性，并提出了一种通用的构造方法。